To find the value of $b$ in the regression line equation $y = a + bx$, we use the formula for the slope of the regression line:

$b = \frac{n\sum(xy) - \sum x \sum y}{n\sum(x^2) - (\sum x)^2}$

Steps to Calculate:

1. Compute the necessary sums:

   • $\sum x$
   • $\sum y$
   • $\sum xy$
   • $\sum x^2$
   • $n$ (number of data points)

2. Plug the values into the formula.

I will calculate this now.import pandas as pd

Given data

x = [15, 17, 19, 20, 20, 21, 22, 22, 23] y = [40, 40, 38, 35, 31, 32, 28, 25, 25]

Create a DataFrame to help with calculations

data = pd.DataFrame({'x': x, 'y': y}) data['xy'] = data['x'] * data['y'] data['x^2'] = data['x'] ** 2

Calculate the necessary sums

n = len(data) # number of data points sum_x = data['x'].sum() sum_y = data['y'].sum() sum_xy = data['xy'].sum() sum_x2 = data['x^2'].sum()

Calculate b

b = (n * sum_xy - sum_x * sum_y) / (n * sum_x2 - sum_x**2) b-2.1239495798319328-2.1239495798319328successThe value of $b$ in the regression line $y = a + bx$ is approximately $-2.12$ (rounded to two decimal places).

Would you like a detailed explanation of how this result was derived or any further assistance?

Here are five related questions to explore further:

1. What is the interpretation of the slope $b$ in this context?
2. How do you calculate the intercept $a$ for the regression line?
3. Can you provide predictions for $y$ using this regression equation?
4. How would the value of $b$ change if one data point were altered?
5. What does the sign of $b$ indicate about the relationship between $x$ and $y$?

Tip: Always visualize your data with a scatterplot to better understand the relationship before calculating regression!